We can solve this problem by listing out the box's individual sides. There are 6 of them here.
Keep in mind that opposite sides have the same areas :D
There's one side, the side measuring 5 in. by 4 in. = 20 square inches. As there are two of those, they amount to a total of 40 square inches.
Then there's a lengthy rectangular side, measuring 12 in. by 4 in. = 48 square inches. Once again, there are two, so those two sides have 96 square inches total.
Then there's the bottom side, measuring 12 in. by 5 in. = 60 square inches. As the top and bottom are identical, those amount to 120 square inches total.
So, the total surface area of the shoebox will be 40 + 96 + 120 = 256 square inches :D